Almost every quasinilpotent Hilbert space operator is a universal quasinilpotent
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 by Domingo A. Herrero PDF
 Proc. Amer. Math. Soc. 71 (1978), 212216 Request permission
Abstract:
Let Q be a quasinilpotent operator acting on a complex separable infinite dimensional Hilbert space; then either ${Q^k}$ is compact for some positive integer k, or the closure of the similarity orbit of Q contains every quasinilpotent operator. Analogous results are shown to be true for the Calkin algebra and for nonseparable Hilbert spaces. For the nonseparable case, the analogous result is true for the closed bilateral ideal $\mathcal {J}$, strictly larger than the ideal of compact operators, if and only if $\mathcal {J}$ is not the ideal associated with an ${\aleph _0}$regular limit cardinal. For the ideal of compact operators, the problem remains open.References

J. Barría and D. A. Herrero, Closure of similarity orbits of nilpotent operators. II (to appear).
 R. G. Douglas and Carl Pearcy, A note on quasitriangular operators, Duke Math. J. 37 (1970), 177–188. MR 257790
 G. Edgar, J. Ernest, and S. G. Lee, Weighing operator spectra, Indiana Univ. Math. J. 21 (1971/72), 61–80. MR 417836, DOI 10.1512/iumj.1971.21.21005
 Carl Pearcy, Ciprian Foiaş, and Dan Voiculescu, Biquasitriangular operators and quasisimilarity, Linear spaces and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977) Lecture Notes in Biomath., vol. 21, Springer, BerlinNew York, 1978, pp. 47–52. MR 501466
 Sandy Grabiner, Nilpotents in Banach algebras, J. London Math. Soc. (2) 14 (1976), no. 1, 7–12. MR 442683, DOI 10.1112/jlms/s214.1.7
 Domingo A. Herrero, Normal limits of nilpotent operators, Indiana Univ. Math. J. 23 (1973/74), 1097–1108. MR 350476, DOI 10.1512/iumj.1974.23.23089
 Domingo A. Herrero, Universal quasinilpotent operators, Acta Sci. Math. (Szeged) 38 (1976), no. 34, 291–300. MR 442728 —, Clausura de las órbitas de similaridad de operadores en espacios de Hilbert, Rev. Un. Mat. Argentina 27 (1976), 244260.
 Domingo A. Herrero, Norm limits of nilpotent operators and weighted spectra in nonseparable Hilbert spaces, Rev. Un. Mat. Argentina 27 (1974/75), no. 2, 83–105. MR 442726
 Catherine L. Olsen, A structure theorem for polynomially compact operators, Amer. J. Math. 93 (1971), 686–698. MR 405152, DOI 10.2307/2373464
 GianCarlo Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469–472. MR 112040, DOI 10.1002/cpa.3160130309
 Shôichirô Sakai, $C^*$algebras and $W^*$algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, SpringerVerlag, New YorkHeidelberg, 1971. MR 0442701
 Joseph G. Stampfli, Compact perturbations, normal eigenvalues and a problem of Salinas, J. London Math. Soc. (2) 9 (1974/75), 165–175. MR 365196, DOI 10.1112/jlms/s29.1.165
 Dan Voiculescu, A noncommutative Weylvon Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338
Additional Information
 © Copyright 1978 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 71 (1978), 212216
 MSC: Primary 47A65; Secondary 47B05
 DOI: https://doi.org/10.1090/S00029939197805004921
 MathSciNet review: 500492